{ "wireMagic": "pb3", "pens": [ { } ], "wsWinStyles": [ { } ], "wpWinPositions": [ { } ], "events": [ { "tStartMs": 0, "dDurationMs": 1260, "segs": [ { "utf8": "This is a Galton board." } ] }, { "tStartMs": 2520, "dDurationMs": 3367, "segs": [ { "utf8": "Maybe you've seen one before, it's a popular demonstration of how, " } ] }, { "tStartMs": 5887, "dDurationMs": 4422, "segs": [ { "utf8": "even when a single event is chaotic and random, with an effectively unknowable outcome, " } ] }, { "tStartMs": 10309, "dDurationMs": 3970, "segs": [ { "utf8": "it's still possible to make precise statements about a large number of events, " } ] }, { "tStartMs": 14279, "dDurationMs": 4021, "segs": [ { "utf8": "namely how the relative proportions for many different outcomes are distributed." } ] }, { "tStartMs": 20380, "dDurationMs": 3822, "segs": [ { "utf8": "More specifically, the Galton board illustrates one of the most prominent " } ] }, { "tStartMs": 24202, "dDurationMs": 3513, "segs": [ { "utf8": "distributions in all probability, known as the normal distribution, " } ] }, { "tStartMs": 27715, "dDurationMs": 4185, "segs": [ { "utf8": "more colloquially known as a bell curve, and also called a Gaussian distribution." } ] }, { "tStartMs": 32500, "dDurationMs": 3865, "segs": [ { "utf8": "There's a very specific function to describe this distribution, it's very pretty, " } ] }, { "tStartMs": 36365, "dDurationMs": 3724, "segs": [ { "utf8": "we'll get into it later, but right now I just want to emphasize how the normal " } ] }, { "tStartMs": 40089, "dDurationMs": 2452, "segs": [ { "utf8": "distribution is, as the name suggests, very common, " } ] }, { "tStartMs": 42541, "dDurationMs": 2499, "segs": [ { "utf8": "it shows up in a lot of seemingly unrelated contexts." } ] }, { "tStartMs": 46020, "dDurationMs": 3558, "segs": [ { "utf8": "If you were to take a large number of people who sit in a similar demographic " } ] }, { "tStartMs": 49578, "dDurationMs": 3422, "segs": [ { "utf8": "and plot their heights, those heights tend to follow a normal distribution." } ] }, { "tStartMs": 53660, "dDurationMs": 3120, "segs": [ { "utf8": "If you look at a large swath of very big natural numbers, " } ] }, { "tStartMs": 56780, "dDurationMs": 4359, "segs": [ { "utf8": "and you ask how many distinct prime factors does each one of those numbers have, " } ] }, { "tStartMs": 61139, "dDurationMs": 3820, "segs": [ { "utf8": "the answers will very closely track with a certain normal distribution." } ] }, { "tStartMs": 65580, "dDurationMs": 4207, "segs": [ { "utf8": "Now our topic for today is one of the crown jewels in all of probability theory, " } ] }, { "tStartMs": 69787, "dDurationMs": 4415, "segs": [ { "utf8": "it's one of the key facts that explains why this distribution is as common as it is, " } ] }, { "tStartMs": 74202, "dDurationMs": 1818, "segs": [ { "utf8": "known as the central limit theorem." } ] }, { "tStartMs": 76640, "dDurationMs": 2121, "segs": [ { "utf8": "This lesson is meant to go back to the basics, " } ] }, { "tStartMs": 78761, "dDurationMs": 3294, "segs": [ { "utf8": "giving you the fundamentals on what the central limit theorem is saying, " } ] }, { "tStartMs": 82055, "dDurationMs": 3205, "segs": [ { "utf8": "what normal distributions are, and I want to assume minimal background." } ] }, { "tStartMs": 85260, "dDurationMs": 4104, "segs": [ { "utf8": "We're going to go decently deep into it, but after this I'd still like to " } ] }, { "tStartMs": 89364, "dDurationMs": 2607, "segs": [ { "utf8": "go deeper and explain why the theorem is true, " } ] }, { "tStartMs": 91971, "dDurationMs": 4104, "segs": [ { "utf8": "why the function underlying the normal distribution has the very specific " } ] }, { "tStartMs": 96075, "dDurationMs": 3716, "segs": [ { "utf8": "form that it does, why that formula has a pi in it, and, most fun, " } ] }, { "tStartMs": 99791, "dDurationMs": 4271, "segs": [ { "utf8": "why those last two facts are actually more related than a lot of traditional " } ] }, { "tStartMs": 104062, "dDurationMs": 1498, "segs": [ { "utf8": "explanations would suggest." } ] }, { "tStartMs": 106480, "dDurationMs": 3567, "segs": [ { "utf8": "That second lesson is also meant to be the follow-on to the convolutions " } ] }, { "tStartMs": 110047, "dDurationMs": 3323, "segs": [ { "utf8": "video that I promised, so there's a lot of interrelated topics here." } ] }, { "tStartMs": 113570, "dDurationMs": 2684, "segs": [ { "utf8": "But right now, back to the fundamentals, I'd like to kick " } ] }, { "tStartMs": 116254, "dDurationMs": 2916, "segs": [ { "utf8": "things off with an overly simplified model of the Galton board." } ] }, { "tStartMs": 120890, "dDurationMs": 4356, "segs": [ { "utf8": "In this model we will assume that each ball falls directly onto a certain central peg, " } ] }, { "tStartMs": 125246, "dDurationMs": 3856, "segs": [ { "utf8": "and that it has a 50-50 probability of bouncing to the left or to the right, " } ] }, { "tStartMs": 129102, "dDurationMs": 4357, "segs": [ { "utf8": "and we'll think of each of those outcomes as either adding one or subtracting one from " } ] }, { "tStartMs": 133459, "dDurationMs": 651, "segs": [ { "utf8": "its position." } ] }, { "tStartMs": 134670, "dDurationMs": 4064, "segs": [ { "utf8": "Once one of those is chosen, we make the highly unrealistic assumption that it " } ] }, { "tStartMs": 138734, "dDurationMs": 3499, "segs": [ { "utf8": "happens to land dead on in the middle of the peg adjacent below it, " } ] }, { "tStartMs": 142233, "dDurationMs": 4168, "segs": [ { "utf8": "where again it'll be faced with the same 50-50 choice of bouncing to the left or " } ] }, { "tStartMs": 146401, "dDurationMs": 669, "segs": [ { "utf8": "to the right." } ] }, { "tStartMs": 147430, "dDurationMs": 3686, "segs": [ { "utf8": "For the one I'm showing on screen, there are five different rows of pegs, " } ] }, { "tStartMs": 151116, "dDurationMs": 3985, "segs": [ { "utf8": "so our little hopping ball makes five different random choices between plus one " } ] }, { "tStartMs": 155101, "dDurationMs": 3835, "segs": [ { "utf8": "and minus one, and we can think of its final position as basically being the " } ] }, { "tStartMs": 158936, "dDurationMs": 3836, "segs": [ { "utf8": "sum of all of those different numbers, which in this case happens to be one, " } ] }, { "tStartMs": 162772, "dDurationMs": 4084, "segs": [ { "utf8": "and we might label all of the different buckets with the sum that they represent, " } ] }, { "tStartMs": 166856, "dDurationMs": 4434, "segs": [ { "utf8": "as we repeat this we're looking at different possible sums for those five random numbers." } ] }, { "tStartMs": 173050, "dDurationMs": 4292, "segs": [ { "utf8": "And for those of you who are inclined to complain that this is a highly unrealistic model " } ] }, { "tStartMs": 177342, "dDurationMs": 4293, "segs": [ { "utf8": "for the true Galton board, let me emphasize the goal right now is not to accurately model " } ] }, { "tStartMs": 181635, "dDurationMs": 4150, "segs": [ { "utf8": "physics, the goal is to give a simple example to illustrate the central limit theorem, " } ] }, { "tStartMs": 185785, "dDurationMs": 4245, "segs": [ { "utf8": "and for that, idealized though this might be, it actually gives us a really good example." } ] }, { "tStartMs": 190570, "dDurationMs": 3170, "segs": [ { "utf8": "If we let many different balls fall, making yet another unrealistic " } ] }, { "tStartMs": 193740, "dDurationMs": 3496, "segs": [ { "utf8": "assumption that they don't influence each other, as if they're all ghosts, " } ] }, { "tStartMs": 197236, "dDurationMs": 3170, "segs": [ { "utf8": "then the number of balls that fall into each different bucket gives " } ] }, { "tStartMs": 200406, "dDurationMs": 2984, "segs": [ { "utf8": "us some loose sense for how likely each one of those buckets is." } ] }, { "tStartMs": 203830, "dDurationMs": 3068, "segs": [ { "utf8": "In this example, the numbers are simple enough that it's not too hard to " } ] }, { "tStartMs": 206898, "dDurationMs": 3112, "segs": [ { "utf8": "explicitly calculate what the probability is for falling into each bucket." } ] }, { "tStartMs": 210270, "dDurationMs": 3748, "segs": [ { "utf8": "If you do want to think that through, you'll find it very reminiscent of Pascal's " } ] }, { "tStartMs": 214018, "dDurationMs": 3840, "segs": [ { "utf8": "triangle, but the neat thing about our theorem is how far it goes beyond the simple " } ] }, { "tStartMs": 217858, "dDurationMs": 412, "segs": [ { "utf8": "examples." } ] }, { "tStartMs": 218670, "dDurationMs": 3013, "segs": [ { "utf8": "So to start off at least, rather than making explicit calculations, " } ] }, { "tStartMs": 221683, "dDurationMs": 3811, "segs": [ { "utf8": "let's just simulate things by running a large number of samples and letting the total " } ] }, { "tStartMs": 225494, "dDurationMs": 3988, "segs": [ { "utf8": "number of results in each different outcome give us some sense for what that distribution " } ] }, { "tStartMs": 229482, "dDurationMs": 488, "segs": [ { "utf8": "looks like." } ] }, { "tStartMs": 230450, "dDurationMs": 2825, "segs": [ { "utf8": "As I said, the one on screen has five rows, so each " } ] }, { "tStartMs": 233275, "dDurationMs": 2935, "segs": [ { "utf8": "sum that we're considering includes only five numbers." } ] }, { "tStartMs": 236810, "dDurationMs": 5091, "segs": [ { "utf8": "The basic idea of the central limit theorem is that if you increase the size of that sum, " } ] }, { "tStartMs": 241901, "dDurationMs": 4243, "segs": [ { "utf8": "for example here would mean increasing the number of rows of pegs for each " } ] }, { "tStartMs": 246144, "dDurationMs": 4074, "segs": [ { "utf8": "ball to bounce off, then the distribution that describes where that sum " } ] }, { "tStartMs": 250218, "dDurationMs": 3112, "segs": [ { "utf8": "is going to fall looks more and more like a bell curve." } ] }, { "tStartMs": 255470, "dDurationMs": 2880, "segs": [ { "utf8": "Here, it's actually worth taking a moment to write down that general idea." } ] }, { "tStartMs": 259269, "dDurationMs": 4378, "segs": [ { "utf8": "The setup is that we have a random variable, and that's basically shorthand for " } ] }, { "tStartMs": 263647, "dDurationMs": 4543, "segs": [ { "utf8": "a random process where each outcome of that process is associated with some number." } ] }, { "tStartMs": 268490, "dDurationMs": 1480, "segs": [ { "utf8": "We'll call that random number x." } ] }, { "tStartMs": 269970, "dDurationMs": 4420, "segs": [ { "utf8": "For example, each bounce off the peg is a random process modeled with two outcomes." } ] }, { "tStartMs": 274850, "dDurationMs": 3040, "segs": [ { "utf8": "Those outcomes are associated with the numbers negative one and positive one." } ] }, { "tStartMs": 278530, "dDurationMs": 2867, "segs": [ { "utf8": "Another example of a random variable would be rolling a die, " } ] }, { "tStartMs": 281397, "dDurationMs": 3433, "segs": [ { "utf8": "where you have six different outcomes, each one associated with a number." } ] }, { "tStartMs": 285470, "dDurationMs": 2272, "segs": [ { "utf8": "What we're doing is taking multiple different " } ] }, { "tStartMs": 287742, "dDurationMs": 2668, "segs": [ { "utf8": "samples of that variable and adding them all together." } ] }, { "tStartMs": 290770, "dDurationMs": 3462, "segs": [ { "utf8": "On our Galton board, that looks like letting the ball bounce off multiple " } ] }, { "tStartMs": 294232, "dDurationMs": 3369, "segs": [ { "utf8": "different pegs on its way down to the bottom, and in the case of a die, " } ] }, { "tStartMs": 297601, "dDurationMs": 3369, "segs": [ { "utf8": "you might imagine rolling many different dice and adding up the results." } ] }, { "tStartMs": 301430, "dDurationMs": 4244, "segs": [ { "utf8": "The claim of the central limit theorem is that as you let the size of that sum " } ] }, { "tStartMs": 305674, "dDurationMs": 3116, "segs": [ { "utf8": "get bigger and bigger, then the distribution of that sum, " } ] }, { "tStartMs": 308790, "dDurationMs": 3063, "segs": [ { "utf8": "how likely it is to fall into different possible values, " } ] }, { "tStartMs": 311853, "dDurationMs": 2257, "segs": [ { "utf8": "will look more and more like a bell curve." } ] }, { "tStartMs": 315430, "dDurationMs": 1700, "segs": [ { "utf8": "That's it, that is the general idea." } ] }, { "tStartMs": 317550, "dDurationMs": 3980, "segs": [ { "utf8": "Over the course of this lesson, our job is to make that statement more quantitative." } ] }, { "tStartMs": 322070, "dDurationMs": 2536, "segs": [ { "utf8": "We're going to put some numbers to it, put some formulas to it, " } ] }, { "tStartMs": 324606, "dDurationMs": 1744, "segs": [ { "utf8": "show how you can use it to make predictions." } ] }, { "tStartMs": 327210, "dDurationMs": 2135, "segs": [ { "utf8": "For example, here's the kind of question I want " } ] }, { "tStartMs": 329345, "dDurationMs": 2225, "segs": [ { "utf8": "you to be able to answer by the end of this video." } ] }, { "tStartMs": 332190, "dDurationMs": 3700, "segs": [ { "utf8": "Suppose you rolled a die 100 times and you added together the results." } ] }, { "tStartMs": 336630, "dDurationMs": 2770, "segs": [ { "utf8": "Could you find a range of values such that you're " } ] }, { "tStartMs": 339400, "dDurationMs": 2770, "segs": [ { "utf8": "95% sure that the sum will fall within that range?" } ] }, { "tStartMs": 342830, "dDurationMs": 3720, "segs": [ { "utf8": "Or maybe I should say find the smallest possible range of values such that this is true." } ] }, { "tStartMs": 347390, "dDurationMs": 2525, "segs": [ { "utf8": "The neat thing is you'll be able to answer this question " } ] }, { "tStartMs": 349915, "dDurationMs": 2215, "segs": [ { "utf8": "whether it's a fair die or if it's a weighted die." } ] }, { "tStartMs": 353450, "dDurationMs": 3296, "segs": [ { "utf8": "Now let me say at the top that this theorem has three different assumptions " } ] }, { "tStartMs": 356746, "dDurationMs": 3384, "segs": [ { "utf8": "that go into it, three things that have to be true before the theorem follows." } ] }, { "tStartMs": 360430, "dDurationMs": 3360, "segs": [ { "utf8": "And I'm actually not going to tell you what they are until the very end of the video." } ] }, { "tStartMs": 364270, "dDurationMs": 2762, "segs": [ { "utf8": "Instead I want you to keep your eye out and see if you can notice " } ] }, { "tStartMs": 367032, "dDurationMs": 2638, "segs": [ { "utf8": "and maybe predict what those three assumptions are going to be." } ] }, { "tStartMs": 370710, "dDurationMs": 3226, "segs": [ { "utf8": "As a next step, to better illustrate just how general this theorem is, " } ] }, { "tStartMs": 373936, "dDurationMs": 3454, "segs": [ { "utf8": "I want to run a couple more simulations for you focused on the dice example." } ] }, { "tStartMs": 380910, "dDurationMs": 3313, "segs": [ { "utf8": "Usually if you think of rolling a die you think of the six outcomes as " } ] }, { "tStartMs": 384223, "dDurationMs": 3407, "segs": [ { "utf8": "being equally probable, but the theorem actually doesn't care about that." } ] }, { "tStartMs": 387830, "dDurationMs": 3386, "segs": [ { "utf8": "We could start with a weighted die, something with a non-trivial " } ] }, { "tStartMs": 391216, "dDurationMs": 3334, "segs": [ { "utf8": "distribution across the outcomes, and the core idea still holds." } ] }, { "tStartMs": 395030, "dDurationMs": 2631, "segs": [ { "utf8": "For the simulation what I'll do is take some distribution " } ] }, { "tStartMs": 397661, "dDurationMs": 2269, "segs": [ { "utf8": "like this one that is skewed towards lower values." } ] }, { "tStartMs": 400250, "dDurationMs": 3622, "segs": [ { "utf8": "I'm going to take 10 distinct samples from that distribution and " } ] }, { "tStartMs": 403872, "dDurationMs": 3678, "segs": [ { "utf8": "then I'll record the sum of that sample on the plot on the bottom." } ] }, { "tStartMs": 408630, "dDurationMs": 3980, "segs": [ { "utf8": "Then I'm going to do this many many different times, always with a sum of size 10, " } ] }, { "tStartMs": 412610, "dDurationMs": 3980, "segs": [ { "utf8": "but keep track of where those sums ended up to give us a sense of the distribution." } ] }, { "tStartMs": 419970, "dDurationMs": 2452, "segs": [ { "utf8": "And in fact let me rescale the y direction to give " } ] }, { "tStartMs": 422422, "dDurationMs": 2308, "segs": [ { "utf8": "us room to run an even larger number of samples." } ] }, { "tStartMs": 425030, "dDurationMs": 2861, "segs": [ { "utf8": "And I'll let it go all the way up to a couple thousand, " } ] }, { "tStartMs": 427891, "dDurationMs": 4599, "segs": [ { "utf8": "and as it does you'll notice that the shape that starts to emerge looks like a bell curve." } ] }, { "tStartMs": 432870, "dDurationMs": 3591, "segs": [ { "utf8": "Maybe if you squint your eyes you can see it skews a tiny bit to the left, " } ] }, { "tStartMs": 436461, "dDurationMs": 4022, "segs": [ { "utf8": "but it's neat that something so symmetric emerged from a starting point that was so " } ] }, { "tStartMs": 440483, "dDurationMs": 527, "segs": [ { "utf8": "asymmetric." } ] }, { "tStartMs": 441470, "dDurationMs": 3267, "segs": [ { "utf8": "To better illustrate what the central limit theorem is all about, " } ] }, { "tStartMs": 444737, "dDurationMs": 2475, "segs": [ { "utf8": "let me run four of these simulations in parallel, " } ] }, { "tStartMs": 447212, "dDurationMs": 4009, "segs": [ { "utf8": "where on the upper left I'm doing it where we're only adding two dice at a time, " } ] }, { "tStartMs": 451221, "dDurationMs": 3664, "segs": [ { "utf8": "on the upper right we're doing it where we're adding five dice at a time, " } ] }, { "tStartMs": 454885, "dDurationMs": 3415, "segs": [ { "utf8": "the lower left is the one that we just saw adding 10 dice at a time, " } ] }, { "tStartMs": 458300, "dDurationMs": 3070, "segs": [ { "utf8": "and then we'll do another one with a bigger sum, 15 at a time." } ] }, { "tStartMs": 462250, "dDurationMs": 2860, "segs": [ { "utf8": "Notice how on the upper left when we're just adding two dice, " } ] }, { "tStartMs": 465110, "dDurationMs": 3044, "segs": [ { "utf8": "the resulting distribution doesn't really look like a bell curve, " } ] }, { "tStartMs": 468154, "dDurationMs": 3876, "segs": [ { "utf8": "it looks a lot more reminiscent of the one we started with, skewed towards the left." } ] }, { "tStartMs": 472810, "dDurationMs": 2618, "segs": [ { "utf8": "But as we allow for more and more dice in each sum, " } ] }, { "tStartMs": 475428, "dDurationMs": 4382, "segs": [ { "utf8": "the resulting shape that comes up in these distributions looks more and more symmetric." } ] }, { "tStartMs": 479950, "dDurationMs": 3940, "segs": [ { "utf8": "It has the lump in the middle and fade towards the tail's shape of a bell curve." } ] }, { "tStartMs": 487050, "dDurationMs": 3440, "segs": [ { "utf8": "And let me emphasize again, you can start with any different distribution." } ] }, { "tStartMs": 490490, "dDurationMs": 3500, "segs": [ { "utf8": "Here I'll run it again, but where most of the probability is tied up " } ] }, { "tStartMs": 493990, "dDurationMs": 3500, "segs": [ { "utf8": "in the numbers 1 and 6, with very low probability for the mid values." } ] }, { "tStartMs": 498190, "dDurationMs": 3934, "segs": [ { "utf8": "Despite completely changing the distribution for an individual roll of the die, " } ] }, { "tStartMs": 502124, "dDurationMs": 4426, "segs": [ { "utf8": "it's still the case that a bell curve shape will emerge as we consider the different sums." } ] }, { "tStartMs": 507270, "dDurationMs": 3114, "segs": [ { "utf8": "Illustrating things with a simulation like this is very fun, " } ] }, { "tStartMs": 510384, "dDurationMs": 2757, "segs": [ { "utf8": "and it's kind of neat to see order emerge from chaos, " } ] }, { "tStartMs": 513141, "dDurationMs": 1889, "segs": [ { "utf8": "but it also feels a little imprecise." } ] }, { "tStartMs": 515390, "dDurationMs": 3169, "segs": [ { "utf8": "Like in this case, when I cut off the simulation at 3000 samples, " } ] }, { "tStartMs": 518559, "dDurationMs": 2306, "segs": [ { "utf8": "even though it kind of looks like a bell curve, " } ] }, { "tStartMs": 520865, "dDurationMs": 3026, "segs": [ { "utf8": "the different buckets seem pretty spiky, and you might wonder, " } ] }, { "tStartMs": 523891, "dDurationMs": 3266, "segs": [ { "utf8": "is it supposed to look that way, or is that just an artifact of the " } ] }, { "tStartMs": 527157, "dDurationMs": 1393, "segs": [ { "utf8": "randomness in the simulation?" } ] }, { "tStartMs": 529010, "dDurationMs": 3141, "segs": [ { "utf8": "And if it is, how many samples do we need before we can be sure that " } ] }, { "tStartMs": 532151, "dDurationMs": 2959, "segs": [ { "utf8": "what we're looking at is representative of the true distribution?" } ] }, { "tStartMs": 539190, "dDurationMs": 3186, "segs": [ { "utf8": "Instead moving forward, let's get a little more theoretical and show " } ] }, { "tStartMs": 542376, "dDurationMs": 3094, "segs": [ { "utf8": "the precise shape these distributions will take on in the long run." } ] }, { "tStartMs": 546130, "dDurationMs": 4237, "segs": [ { "utf8": "The easiest case to make this calculation is if we have a uniform distribution, " } ] }, { "tStartMs": 550367, "dDurationMs": 3603, "segs": [ { "utf8": "where each possible face of the die has an equal probability, 1 6th." } ] }, { "tStartMs": 553990, "dDurationMs": 4281, "segs": [ { "utf8": "For example, if you then want to know how likely different sums are for a pair of dice, " } ] }, { "tStartMs": 558271, "dDurationMs": 3455, "segs": [ { "utf8": "it's essentially a counting game, where you count up how many distinct " } ] }, { "tStartMs": 561726, "dDurationMs": 2968, "segs": [ { "utf8": "pairs take on the same sum, which in the diagram I've drawn, " } ] }, { "tStartMs": 564694, "dDurationMs": 3796, "segs": [ { "utf8": "you can conveniently think about by going through all the different diagonals." } ] }, { "tStartMs": 571410, "dDurationMs": 2856, "segs": [ { "utf8": "Since each such pair has an equal chance of showing up, " } ] }, { "tStartMs": 574266, "dDurationMs": 3264, "segs": [ { "utf8": "1 in 36, all you have to do is count the sizes of these buckets." } ] }, { "tStartMs": 578190, "dDurationMs": 4238, "segs": [ { "utf8": "That gives us a definitive shape for the distribution describing a sum of two dice, " } ] }, { "tStartMs": 582428, "dDurationMs": 3280, "segs": [ { "utf8": "and if we were to play the same game with all possible triplets, " } ] }, { "tStartMs": 585708, "dDurationMs": 2422, "segs": [ { "utf8": "the resulting distribution would look like this." } ] }, { "tStartMs": 588690, "dDurationMs": 2602, "segs": [ { "utf8": "Now what's more challenging, but a lot more interesting, " } ] }, { "tStartMs": 591292, "dDurationMs": 3698, "segs": [ { "utf8": "is to ask what happens if we have a non-uniform distribution for that single die." } ] }, { "tStartMs": 595550, "dDurationMs": 2420, "segs": [ { "utf8": "We actually talked all about this in the last video." } ] }, { "tStartMs": 598450, "dDurationMs": 2516, "segs": [ { "utf8": "You do essentially the same thing, you go through all " } ] }, { "tStartMs": 600966, "dDurationMs": 2704, "segs": [ { "utf8": "the distinct pairs of dice which add up to the same value." } ] }, { "tStartMs": 603970, "dDurationMs": 2508, "segs": [ { "utf8": "It's just that instead of counting those pairs, " } ] }, { "tStartMs": 606478, "dDurationMs": 4390, "segs": [ { "utf8": "for each pair you multiply the two probabilities of each particular face coming up, " } ] }, { "tStartMs": 610868, "dDurationMs": 1882, "segs": [ { "utf8": "and then you add all those together." } ] }, { "tStartMs": 613290, "dDurationMs": 3392, "segs": [ { "utf8": "The computation that does this for all possible sums has a fancy name, " } ] }, { "tStartMs": 616682, "dDurationMs": 3679, "segs": [ { "utf8": "it's called a convolution, but it's essentially just the weighted version of " } ] }, { "tStartMs": 620361, "dDurationMs": 4109, "segs": [ { "utf8": "the counting game that anyone who's played with a pair of dice already finds familiar." } ] }, { "tStartMs": 625030, "dDurationMs": 3914, "segs": [ { "utf8": "For our purposes in this lesson, I'll have the computer calculate all that, " } ] }, { "tStartMs": 628944, "dDurationMs": 4120, "segs": [ { "utf8": "simply display the results for you, and invite you to observe certain patterns, " } ] }, { "tStartMs": 633064, "dDurationMs": 2266, "segs": [ { "utf8": "but under the hood, this is what's going on." } ] }, { "tStartMs": 636650, "dDurationMs": 3274, "segs": [ { "utf8": "So just to be crystal clear on what's being represented here, " } ] }, { "tStartMs": 639924, "dDurationMs": 3855, "segs": [ { "utf8": "if you imagine sampling two different values from that top distribution, " } ] }, { "tStartMs": 643779, "dDurationMs": 3116, "segs": [ { "utf8": "the one describing a single die, and adding them together, " } ] }, { "tStartMs": 646895, "dDurationMs": 3909, "segs": [ { "utf8": "then the second distribution I'm drawing represents how likely you are to " } ] }, { "tStartMs": 650804, "dDurationMs": 1426, "segs": [ { "utf8": "see various different sums." } ] }, { "tStartMs": 652890, "dDurationMs": 4178, "segs": [ { "utf8": "Likewise, if you imagine sampling three distinct values from that top distribution, " } ] }, { "tStartMs": 657068, "dDurationMs": 3432, "segs": [ { "utf8": "and adding them together, the next plot represents the probabilities " } ] }, { "tStartMs": 660500, "dDurationMs": 1990, "segs": [ { "utf8": "for various different sums in that case." } ] }, { "tStartMs": 663510, "dDurationMs": 4492, "segs": [ { "utf8": "So if I compute what the distributions for these sums look like for larger and larger " } ] }, { "tStartMs": 668002, "dDurationMs": 4388, "segs": [ { "utf8": "sums, well you know what I'm going to say, it looks more and more like a bell curve." } ] }, { "tStartMs": 673350, "dDurationMs": 3100, "segs": [ { "utf8": "But before we get to that, I want you to make a couple more simple observations." } ] }, { "tStartMs": 677450, "dDurationMs": 3442, "segs": [ { "utf8": "For example, these distributions seem to be wandering to the right, " } ] }, { "tStartMs": 680892, "dDurationMs": 3898, "segs": [ { "utf8": "and also they seem to be getting more spread out, and a little bit more flat." } ] }, { "tStartMs": 685250, "dDurationMs": 2661, "segs": [ { "utf8": "You cannot describe the central limit theorem quantitatively " } ] }, { "tStartMs": 687911, "dDurationMs": 2225, "segs": [ { "utf8": "without taking into account both of those effects, " } ] }, { "tStartMs": 690136, "dDurationMs": 3054, "segs": [ { "utf8": "which in turn requires describing the mean and the standard deviation." } ] }, { "tStartMs": 693950, "dDurationMs": 3714, "segs": [ { "utf8": "Maybe you're already familiar with those, but I want to make minimal assumptions here, " } ] }, { "tStartMs": 697664, "dDurationMs": 2946, "segs": [ { "utf8": "and it never hurts to review, so let's quickly go over both of those." } ] }, { "tStartMs": 703410, "dDurationMs": 3789, "segs": [ { "utf8": "The mean of a distribution, often denoted with the Greek letter mu, " } ] }, { "tStartMs": 707199, "dDurationMs": 3511, "segs": [ { "utf8": "is a way of capturing the center of mass for that distribution." } ] }, { "tStartMs": 711190, "dDurationMs": 3241, "segs": [ { "utf8": "It's calculated as the expected value of our random variable, " } ] }, { "tStartMs": 714431, "dDurationMs": 4183, "segs": [ { "utf8": "which is a way of saying you go through all of the different possible outcomes, " } ] }, { "tStartMs": 718614, "dDurationMs": 4236, "segs": [ { "utf8": "and you multiply the probability of that outcome times the value of the variable." } ] }, { "tStartMs": 723190, "dDurationMs": 3220, "segs": [ { "utf8": "If higher values are more probable, that weighted sum is going to be bigger." } ] }, { "tStartMs": 726750, "dDurationMs": 3200, "segs": [ { "utf8": "If lower values are more probable, that weighted sum is going to be smaller." } ] }, { "tStartMs": 730790, "dDurationMs": 3891, "segs": [ { "utf8": "A little more interesting is if you want to measure how spread out this distribution is, " } ] }, { "tStartMs": 734681, "dDurationMs": 2449, "segs": [ { "utf8": "because there's multiple different ways you might do it." } ] }, { "tStartMs": 738530, "dDurationMs": 1760, "segs": [ { "utf8": "One of them is called the variance." } ] }, { "tStartMs": 740830, "dDurationMs": 4537, "segs": [ { "utf8": "The idea there is to look at the difference between each possible value and the mean, " } ] }, { "tStartMs": 745367, "dDurationMs": 2903, "segs": [ { "utf8": "square that difference, and ask for its expected value." } ] }, { "tStartMs": 748730, "dDurationMs": 2847, "segs": [ { "utf8": "The idea is that whether your value is below or above the mean, " } ] }, { "tStartMs": 751577, "dDurationMs": 2670, "segs": [ { "utf8": "when you square that difference, you get a positive number, " } ] }, { "tStartMs": 754247, "dDurationMs": 2403, "segs": [ { "utf8": "and the larger the difference, the bigger that number." } ] }, { "tStartMs": 757370, "dDurationMs": 3795, "segs": [ { "utf8": "Squaring it like this turns out to make the math much much nicer than if we did " } ] }, { "tStartMs": 761165, "dDurationMs": 3985, "segs": [ { "utf8": "something like an absolute value, but the downside is that it's hard to think about " } ] }, { "tStartMs": 765150, "dDurationMs": 2894, "segs": [ { "utf8": "this as a distance in our diagram because the units are off, " } ] }, { "tStartMs": 768044, "dDurationMs": 4079, "segs": [ { "utf8": "kind of like the units here are square units, whereas a distance in our diagram would " } ] }, { "tStartMs": 772123, "dDurationMs": 1187, "segs": [ { "utf8": "be a kind of linear unit." } ] }, { "tStartMs": 773710, "dDurationMs": 3588, "segs": [ { "utf8": "So another way to measure spread is what's called the standard deviation, " } ] }, { "tStartMs": 777298, "dDurationMs": 1892, "segs": [ { "utf8": "which is the square root of this value." } ] }, { "tStartMs": 779470, "dDurationMs": 3856, "segs": [ { "utf8": "That can be interpreted much more reasonably as a distance on our diagram, " } ] }, { "tStartMs": 783326, "dDurationMs": 2827, "segs": [ { "utf8": "and it's commonly denoted with the Greek letter sigma, " } ] }, { "tStartMs": 786153, "dDurationMs": 3497, "segs": [ { "utf8": "so you know m for mean as for standard deviation, but both in Greek." } ] }, { "tStartMs": 791870, "dDurationMs": 2095, "segs": [ { "utf8": "Looking back at our sequence of distributions, " } ] }, { "tStartMs": 793965, "dDurationMs": 2185, "segs": [ { "utf8": "let's talk about the mean and standard deviation." } ] }, { "tStartMs": 796630, "dDurationMs": 2665, "segs": [ { "utf8": "If we call the mean of the initial distribution mu, " } ] }, { "tStartMs": 799295, "dDurationMs": 2564, "segs": [ { "utf8": "which for the one illustrated happens to be 2.24, " } ] }, { "tStartMs": 801859, "dDurationMs": 3332, "segs": [ { "utf8": "hopefully it won't be too surprising if I tell you that the mean " } ] }, { "tStartMs": 805191, "dDurationMs": 1539, "segs": [ { "utf8": "of the next one is 2 times mu." } ] }, { "tStartMs": 807130, "dDurationMs": 3501, "segs": [ { "utf8": "That is, you roll a pair of dice, you want to know the expected value of the sum, " } ] }, { "tStartMs": 810631, "dDurationMs": 2179, "segs": [ { "utf8": "it's two times the expected value for a single die." } ] }, { "tStartMs": 813850, "dDurationMs": 5560, "segs": [ { "utf8": "Similarly, the expected value for our sum of size 3 is 3 times mu, and so on and so forth." } ] }, { "tStartMs": 819630, "dDurationMs": 2078, "segs": [ { "utf8": "The mean just marches steadily on to the right, " } ] }, { "tStartMs": 821708, "dDurationMs": 3162, "segs": [ { "utf8": "which is why our distributions seem to be drifting off in that direction." } ] }, { "tStartMs": 825350, "dDurationMs": 2209, "segs": [ { "utf8": "A little more challenging, but very important, " } ] }, { "tStartMs": 827559, "dDurationMs": 2351, "segs": [ { "utf8": "is to describe how the standard deviation changes." } ] }, { "tStartMs": 830490, "dDurationMs": 3415, "segs": [ { "utf8": "The key fact here is that if you have two different random variables, " } ] }, { "tStartMs": 833905, "dDurationMs": 2976, "segs": [ { "utf8": "then the variance for the sum of those variables is the same " } ] }, { "tStartMs": 836881, "dDurationMs": 2489, "segs": [ { "utf8": "as just adding together the original two variances." } ] }, { "tStartMs": 839930, "dDurationMs": 3700, "segs": [ { "utf8": "This is one of those facts that you can just compute when you unpack all the definitions." } ] }, { "tStartMs": 843630, "dDurationMs": 2580, "segs": [ { "utf8": "There are a couple nice intuitions for why it's true." } ] }, { "tStartMs": 846630, "dDurationMs": 3403, "segs": [ { "utf8": "My tentative plan is to just actually make a series about probability and " } ] }, { "tStartMs": 850033, "dDurationMs": 3497, "segs": [ { "utf8": "talk about things like intuitions underlying variance and its cousins there." } ] }, { "tStartMs": 854010, "dDurationMs": 4186, "segs": [ { "utf8": "But right now, the main thing I want you to highlight is how it's the variance that adds, " } ] }, { "tStartMs": 858196, "dDurationMs": 1954, "segs": [ { "utf8": "it's not the standard deviation that adds." } ] }, { "tStartMs": 860410, "dDurationMs": 4516, "segs": [ { "utf8": "So, critically, if you were to take n different realizations of the same random " } ] }, { "tStartMs": 864926, "dDurationMs": 4347, "segs": [ { "utf8": "variable and ask what the sum looks like, the variance of sum is n times the " } ] }, { "tStartMs": 869273, "dDurationMs": 3839, "segs": [ { "utf8": "variance of your original variable, meaning the standard deviation, " } ] }, { "tStartMs": 873112, "dDurationMs": 4573, "segs": [ { "utf8": "the square root of all this, is the square root of n times the original standard " } ] }, { "tStartMs": 877685, "dDurationMs": 565, "segs": [ { "utf8": "deviation." } ] }, { "tStartMs": 879290, "dDurationMs": 2625, "segs": [ { "utf8": "For example, back in our sequence of distributions, " } ] }, { "tStartMs": 881915, "dDurationMs": 3333, "segs": [ { "utf8": "if we label the standard deviation of our initial one with sigma, " } ] }, { "tStartMs": 885248, "dDurationMs": 4140, "segs": [ { "utf8": "then the next standard deviation is going to be the square root of 2 times sigma, " } ] }, { "tStartMs": 889388, "dDurationMs": 4393, "segs": [ { "utf8": "and after that it looks like the square root of 3 times sigma, and so on and so forth. " } ] }, { "tStartMs": 893781, "dDurationMs": 1869, "segs": [ { "utf8": "This, like I said, is very important." } ] }, { "tStartMs": 896070, "dDurationMs": 2978, "segs": [ { "utf8": "It means that even though our distributions are getting spread out, " } ] }, { "tStartMs": 899048, "dDurationMs": 2628, "segs": [ { "utf8": "they're not spreading out all that quickly, they only do so " } ] }, { "tStartMs": 901676, "dDurationMs": 2454, "segs": [ { "utf8": "in proportion to the square root of the size of the sum." } ] }, { "tStartMs": 904710, "dDurationMs": 4084, "segs": [ { "utf8": "As we prepare to make a more quantitative description of the central limit theorem, " } ] }, { "tStartMs": 908794, "dDurationMs": 4036, "segs": [ { "utf8": "the core intuition I want you to keep in your head is that we'll basically realign " } ] }, { "tStartMs": 912830, "dDurationMs": 3160, "segs": [ { "utf8": "all of these distributions so that their means line up together, " } ] }, { "tStartMs": 915990, "dDurationMs": 3987, "segs": [ { "utf8": "and then rescale them so that all of the standard deviations are just going to be " } ] }, { "tStartMs": 919977, "dDurationMs": 633, "segs": [ { "utf8": "equal to one." } ] }, { "tStartMs": 921290, "dDurationMs": 4437, "segs": [ { "utf8": "And when we do that, the shape that results gets closer and closer to a certain universal " } ] }, { "tStartMs": 925727, "dDurationMs": 4143, "segs": [ { "utf8": "shape, described with an elegant little function that we'll unpack in just a moment." } ] }, { "tStartMs": 930470, "dDurationMs": 4381, "segs": [ { "utf8": "And let me say one more time, the real magic here is how we could have started with " } ] }, { "tStartMs": 934851, "dDurationMs": 4433, "segs": [ { "utf8": "any distribution, describing a single roll of the die, and if we play the same game, " } ] }, { "tStartMs": 939284, "dDurationMs": 3860, "segs": [ { "utf8": "considering what the distributions for the many different sums look like, " } ] }, { "tStartMs": 943144, "dDurationMs": 2451, "segs": [ { "utf8": "and we realign them so that the means line up, " } ] }, { "tStartMs": 945595, "dDurationMs": 3391, "segs": [ { "utf8": "and we rescale them so that the standard deviations are all one, " } ] }, { "tStartMs": 948986, "dDurationMs": 3964, "segs": [ { "utf8": "we still approach that same universal shape, which is kind of mind-boggling." } ] }, { "tStartMs": 954810, "dDurationMs": 2939, "segs": [ { "utf8": "And now, my friends, is probably as good a time as any " } ] }, { "tStartMs": 957749, "dDurationMs": 3101, "segs": [ { "utf8": "to finally get into the formula for a normal distribution." } ] }, { "tStartMs": 961490, "dDurationMs": 2158, "segs": [ { "utf8": "And the way I'd like to do this is to basically peel " } ] }, { "tStartMs": 963648, "dDurationMs": 2282, "segs": [ { "utf8": "back all the layers and build it up one piece at a time." } ] }, { "tStartMs": 966530, "dDurationMs": 4080, "segs": [ { "utf8": "The function e to the x, or anything to the x, describes exponential growth, " } ] }, { "tStartMs": 970610, "dDurationMs": 4398, "segs": [ { "utf8": "and if you make that exponent negative, which flips around the graph horizontally, " } ] }, { "tStartMs": 975008, "dDurationMs": 2862, "segs": [ { "utf8": "you might think of it as describing exponential decay." } ] }, { "tStartMs": 978510, "dDurationMs": 3416, "segs": [ { "utf8": "To make this decay in both directions, you could do something to make sure the " } ] }, { "tStartMs": 981926, "dDurationMs": 3504, "segs": [ { "utf8": "exponent is always negative and growing, like taking the negative absolute value." } ] }, { "tStartMs": 985930, "dDurationMs": 3167, "segs": [ { "utf8": "That would give us this kind of awkward sharp point in the middle, " } ] }, { "tStartMs": 989097, "dDurationMs": 3025, "segs": [ { "utf8": "but if instead you make that exponent the negative square of x, " } ] }, { "tStartMs": 992122, "dDurationMs": 3688, "segs": [ { "utf8": "you get a smoother version of the same thing, which decays in both directions." } ] }, { "tStartMs": 996330, "dDurationMs": 1860, "segs": [ { "utf8": "This gives us the basic bell curve shape." } ] }, { "tStartMs": 998650, "dDurationMs": 2356, "segs": [ { "utf8": "Now if you throw a constant in front of that x, " } ] }, { "tStartMs": 1001006, "dDurationMs": 3191, "segs": [ { "utf8": "and you scale that constant up and down, it lets you stretch and " } ] }, { "tStartMs": 1004197, "dDurationMs": 4173, "segs": [ { "utf8": "squish the graph horizontally, allowing you to describe narrow and wider bell curves." } ] }, { "tStartMs": 1009010, "dDurationMs": 3424, "segs": [ { "utf8": "And a quick thing I'd like to point out here is that based on the " } ] }, { "tStartMs": 1012434, "dDurationMs": 3165, "segs": [ { "utf8": "rules of exponentiation, as we tweak around that constant c, " } ] }, { "tStartMs": 1015599, "dDurationMs": 4151, "segs": [ { "utf8": "you could also think about it as simply changing the base of the exponentiation." } ] }, { "tStartMs": 1020150, "dDurationMs": 3480, "segs": [ { "utf8": "And in that sense, the number e is not really all that special for our formula." } ] }, { "tStartMs": 1024050, "dDurationMs": 2874, "segs": [ { "utf8": "We could replace it with any other positive constant, " } ] }, { "tStartMs": 1026924, "dDurationMs": 3566, "segs": [ { "utf8": "and you'll get the same family of curves as we tweak that constant." } ] }, { "tStartMs": 1031510, "dDurationMs": 1599, "segs": [ { "utf8": "Make it a 2, same family of curves." } ] }, { "tStartMs": 1033329, "dDurationMs": 1740, "segs": [ { "utf8": "Make it a 3, same family of curves." } ] }, { "tStartMs": 1035750, "dDurationMs": 3740, "segs": [ { "utf8": "The reason we use e is that it gives that constant a very readable meaning." } ] }, { "tStartMs": 1040109, "dDurationMs": 4206, "segs": [ { "utf8": "Or rather, if we reconfigure things a little bit so that the exponent looks " } ] }, { "tStartMs": 1044315, "dDurationMs": 3321, "segs": [ { "utf8": "like negative 1 half times x divided by a certain constant, " } ] }, { "tStartMs": 1047636, "dDurationMs": 4150, "segs": [ { "utf8": "which we'll suggestively call sigma squared, then once we turn this into a " } ] }, { "tStartMs": 1051786, "dDurationMs": 4261, "segs": [ { "utf8": "probability distribution, that constant sigma will be the standard deviation " } ] }, { "tStartMs": 1056047, "dDurationMs": 1163, "segs": [ { "utf8": "of that distribution." } ] }, { "tStartMs": 1057810, "dDurationMs": 760, "segs": [ { "utf8": "And that's very nice." } ] }, { "tStartMs": 1058910, "dDurationMs": 3291, "segs": [ { "utf8": "But before we can interpret this as a probability distribution, " } ] }, { "tStartMs": 1062201, "dDurationMs": 2109, "segs": [ { "utf8": "we need the area under the curve to be 1." } ] }, { "tStartMs": 1064830, "dDurationMs": 2080, "segs": [ { "utf8": "And the reason for that is how the curve is interpreted." } ] }, { "tStartMs": 1067370, "dDurationMs": 3281, "segs": [ { "utf8": "Unlike discrete distributions, when it comes to something continuous, " } ] }, { "tStartMs": 1070651, "dDurationMs": 2719, "segs": [ { "utf8": "you don't ask about the probability of a particular point." } ] }, { "tStartMs": 1073790, "dDurationMs": 4440, "segs": [ { "utf8": "Instead, you ask for the probability that a value falls between two different values." } ] }, { "tStartMs": 1078750, "dDurationMs": 3397, "segs": [ { "utf8": "And what the curve is telling you is that that probability " } ] }, { "tStartMs": 1082147, "dDurationMs": 3283, "segs": [ { "utf8": "equals the area under the curve between those two values." } ] }, { "tStartMs": 1086030, "dDurationMs": 3400, "segs": [ { "utf8": "There's a whole other video about this, they're called probability density functions." } ] }, { "tStartMs": 1089830, "dDurationMs": 3917, "segs": [ { "utf8": "The main point right now is that the area under the entire curve represents " } ] }, { "tStartMs": 1093747, "dDurationMs": 3403, "segs": [ { "utf8": "the probability that something happens, that some number comes up." } ] }, { "tStartMs": 1097410, "dDurationMs": 3220, "segs": [ { "utf8": "That should be 1, which is why we want the area under this to be 1." } ] }, { "tStartMs": 1101050, "dDurationMs": 3931, "segs": [ { "utf8": "As it stands with the basic bell curve shape of e to the negative x squared, " } ] }, { "tStartMs": 1104981, "dDurationMs": 2809, "segs": [ { "utf8": "the area is not 1, it's actually the square root of pi." } ] }, { "tStartMs": 1108410, "dDurationMs": 740, "segs": [ { "utf8": "I know, right?" } ] }, { "tStartMs": 1109270, "dDurationMs": 920, "segs": [ { "utf8": "What is pi doing here?" } ] }, { "tStartMs": 1110290, "dDurationMs": 1180, "segs": [ { "utf8": "What does this have to do with circles?" } ] }, { "tStartMs": 1112010, "dDurationMs": 3040, "segs": [ { "utf8": "Like I said at the start, I'd love to talk all about that in the next video." } ] }, { "tStartMs": 1115330, "dDurationMs": 2923, "segs": [ { "utf8": "But if you can spare your excitement, for our purposes right now, " } ] }, { "tStartMs": 1118253, "dDurationMs": 3455, "segs": [ { "utf8": "all it means is that we should divide this function by the square root of pi, " } ] }, { "tStartMs": 1121708, "dDurationMs": 1462, "segs": [ { "utf8": "and it gives us the area we want." } ] }, { "tStartMs": 1123610, "dDurationMs": 3651, "segs": [ { "utf8": "Throwing back in the constants we had earlier, the one half and the sigma, " } ] }, { "tStartMs": 1127261, "dDurationMs": 4042, "segs": [ { "utf8": "the effect there is to stretch out the graph by a factor of sigma times the square " } ] }, { "tStartMs": 1131303, "dDurationMs": 487, "segs": [ { "utf8": "root of 2." } ] }, { "tStartMs": 1132410, "dDurationMs": 4070, "segs": [ { "utf8": "So we also need to divide out by that in order to make sure it has an area of 1, " } ] }, { "tStartMs": 1136480, "dDurationMs": 3167, "segs": [ { "utf8": "and combining those fractions, the factor out front looks like " } ] }, { "tStartMs": 1139647, "dDurationMs": 2463, "segs": [ { "utf8": "1 divided by sigma times the square root of 2 pi." } ] }, { "tStartMs": 1142910, "dDurationMs": 2940, "segs": [ { "utf8": "This, finally, is a valid probability distribution." } ] }, { "tStartMs": 1146450, "dDurationMs": 3986, "segs": [ { "utf8": "As we tweak that value sigma, resulting in narrower and wider curves, " } ] }, { "tStartMs": 1150436, "dDurationMs": 3874, "segs": [ { "utf8": "that constant in the front always guarantees that the area equals 1." } ] }, { "tStartMs": 1155910, "dDurationMs": 2899, "segs": [ { "utf8": "The special case where sigma equals 1 has a specific name, " } ] }, { "tStartMs": 1158809, "dDurationMs": 4226, "segs": [ { "utf8": "we call it the standard normal distribution, which plays an especially important role " } ] }, { "tStartMs": 1163035, "dDurationMs": 1475, "segs": [ { "utf8": "for you and me in this lesson." } ] }, { "tStartMs": 1165130, "dDurationMs": 4808, "segs": [ { "utf8": "And all possible normal distributions are not only parameterized with this value sigma, " } ] }, { "tStartMs": 1169938, "dDurationMs": 3606, "segs": [ { "utf8": "but we also subtract off another constant mu from the variable x, " } ] }, { "tStartMs": 1173544, "dDurationMs": 3770, "segs": [ { "utf8": "and this essentially just lets you slide the graph left and right so " } ] }, { "tStartMs": 1177314, "dDurationMs": 2896, "segs": [ { "utf8": "that you can prescribe the mean of this distribution." } ] }, { "tStartMs": 1180990, "dDurationMs": 2905, "segs": [ { "utf8": "So in short, we have two parameters, one describing the mean, " } ] }, { "tStartMs": 1183895, "dDurationMs": 4170, "segs": [ { "utf8": "one describing the standard deviation, and they're all tied together in this big formula " } ] }, { "tStartMs": 1188065, "dDurationMs": 1125, "segs": [ { "utf8": "involving an e and a pi." } ] }, { "tStartMs": 1189190, "dDurationMs": 5251, "segs": [ { "utf8": "Now that all of that is on the table, let's look back again at the idea of starting with " } ] }, { "tStartMs": 1194441, "dDurationMs": 5074, "segs": [ { "utf8": "some random variable and asking what the distributions for sums of that variable look " } ] }, { "tStartMs": 1199515, "dDurationMs": 295, "segs": [ { "utf8": "like." } ] }, { "tStartMs": 1200130, "dDurationMs": 3242, "segs": [ { "utf8": "As we've already gone over, when you increase the size of that sum, " } ] }, { "tStartMs": 1203372, "dDurationMs": 3195, "segs": [ { "utf8": "the resulting distribution will shift according to a growing mean, " } ] }, { "tStartMs": 1206567, "dDurationMs": 3243, "segs": [ { "utf8": "and it slowly spreads out according to a growing standard deviation." } ] }, { "tStartMs": 1210330, "dDurationMs": 4150, "segs": [ { "utf8": "And putting some actual formulas to it, if we know the mean of our underlying " } ] }, { "tStartMs": 1214480, "dDurationMs": 3884, "segs": [ { "utf8": "random variable, we call it mu, and we also know its standard deviation, " } ] }, { "tStartMs": 1218364, "dDurationMs": 4204, "segs": [ { "utf8": "and we call it sigma, then the mean for the sum on the bottom will be mu times " } ] }, { "tStartMs": 1222568, "dDurationMs": 4204, "segs": [ { "utf8": "the size of the sum, and the standard deviation will be sigma times the square " } ] }, { "tStartMs": 1226772, "dDurationMs": 958, "segs": [ { "utf8": "root of that size." } ] }, { "tStartMs": 1228190, "dDurationMs": 3702, "segs": [ { "utf8": "So now, if we want to claim that this looks more and more like a bell curve, " } ] }, { "tStartMs": 1231892, "dDurationMs": 3077, "segs": [ { "utf8": "and a bell curve is only described by two different parameters, " } ] }, { "tStartMs": 1234969, "dDurationMs": 2741, "segs": [ { "utf8": "the mean and the standard deviation, you know what to do." } ] }, { "tStartMs": 1237930, "dDurationMs": 4452, "segs": [ { "utf8": "You could plug those two values into the formula, and it gives you a highly explicit, " } ] }, { "tStartMs": 1242382, "dDurationMs": 4608, "segs": [ { "utf8": "albeit kind of complicated, formula for a curve that should closely fit our distribution." } ] }, { "tStartMs": 1248390, "dDurationMs": 3066, "segs": [ { "utf8": "But there's another way we can describe it that's a little more " } ] }, { "tStartMs": 1251456, "dDurationMs": 3354, "segs": [ { "utf8": "elegant and lends itself to a very fun visual that we can build up to." } ] }, { "tStartMs": 1255270, "dDurationMs": 3306, "segs": [ { "utf8": "Instead of focusing on the sum of all of these random variables, " } ] }, { "tStartMs": 1258576, "dDurationMs": 3916, "segs": [ { "utf8": "let's modify this expression a little bit, where what we'll do is we'll look " } ] }, { "tStartMs": 1262492, "dDurationMs": 3866, "segs": [ { "utf8": "at the mean that we expect that sum to take, and we subtract it off so that " } ] }, { "tStartMs": 1266358, "dDurationMs": 3815, "segs": [ { "utf8": "our new expression has a mean of zero, and then we're going to look at the " } ] }, { "tStartMs": 1270173, "dDurationMs": 3306, "segs": [ { "utf8": "standard deviation we expect of our sum, and divide out by that, " } ] }, { "tStartMs": 1273479, "dDurationMs": 3968, "segs": [ { "utf8": "which basically just rescales the units so that the standard deviation of our " } ] }, { "tStartMs": 1277447, "dDurationMs": 1323, "segs": [ { "utf8": "expression will equal one." } ] }, { "tStartMs": 1279350, "dDurationMs": 2515, "segs": [ { "utf8": "This might seem like a more complicated expression, " } ] }, { "tStartMs": 1281865, "dDurationMs": 2225, "segs": [ { "utf8": "but it actually has a highly readable meaning." } ] }, { "tStartMs": 1284450, "dDurationMs": 5220, "segs": [ { "utf8": "It's essentially saying how many standard deviations away from the mean is this sum?" } ] }, { "tStartMs": 1290750, "dDurationMs": 4227, "segs": [ { "utf8": "For example, this bar here corresponds to a certain value that you might find when you " } ] }, { "tStartMs": 1294977, "dDurationMs": 4179, "segs": [ { "utf8": "roll 10 dice and you add them all up, and its position a little above negative one is " } ] }, { "tStartMs": 1299156, "dDurationMs": 4276, "segs": [ { "utf8": "telling you that that value is a little bit less than one standard deviation lower than " } ] }, { "tStartMs": 1303432, "dDurationMs": 438, "segs": [ { "utf8": "the mean." } ] }, { "tStartMs": 1305130, "dDurationMs": 3767, "segs": [ { "utf8": "Also, by the way, in anticipation for the animation I'm trying to build to here, " } ] }, { "tStartMs": 1308897, "dDurationMs": 3860, "segs": [ { "utf8": "the way I'm representing things on that lower plot is that the area of each one of " } ] }, { "tStartMs": 1312757, "dDurationMs": 3907, "segs": [ { "utf8": "these bars is telling us the probability of the corresponding value rather than the " } ] }, { "tStartMs": 1316664, "dDurationMs": 326, "segs": [ { "utf8": "height." } ] }, { "tStartMs": 1317230, "dDurationMs": 2252, "segs": [ { "utf8": "You might think of the y-axis as representing " } ] }, { "tStartMs": 1319482, "dDurationMs": 2448, "segs": [ { "utf8": "not probability but a kind of probability density." } ] }, { "tStartMs": 1322270, "dDurationMs": 3726, "segs": [ { "utf8": "The reason for this is to set the stage so that it aligns with the way we " } ] }, { "tStartMs": 1325996, "dDurationMs": 3877, "segs": [ { "utf8": "interpret continuous distributions, where the probability of falling between " } ] }, { "tStartMs": 1329873, "dDurationMs": 3677, "segs": [ { "utf8": "a range of values is equal to an area under a curve between those values." } ] }, { "tStartMs": 1333910, "dDurationMs": 2820, "segs": [ { "utf8": "In particular, the area of all the bars together is going to be one." } ] }, { "tStartMs": 1338230, "dDurationMs": 2720, "segs": [ { "utf8": "Now, with all of that in place, let's have a little fun." } ] }, { "tStartMs": 1341330, "dDurationMs": 4027, "segs": [ { "utf8": "Let me start by rolling things back so that the distribution on the bottom represents " } ] }, { "tStartMs": 1345357, "dDurationMs": 3653, "segs": [ { "utf8": "a relatively small sum, like adding together only three such random variables." } ] }, { "tStartMs": 1349450, "dDurationMs": 2980, "segs": [ { "utf8": "Notice what happens as I change the distribution we start with." } ] }, { "tStartMs": 1352730, "dDurationMs": 3560, "segs": [ { "utf8": "As it changes, the distribution on the bottom completely changes its shape." } ] }, { "tStartMs": 1356510, "dDurationMs": 2260, "segs": [ { "utf8": "It's very dependent on what we started with." } ] }, { "tStartMs": 1360350, "dDurationMs": 3828, "segs": [ { "utf8": "If we let the size of our sum get a little bit bigger, say going up to 10, " } ] }, { "tStartMs": 1364178, "dDurationMs": 4287, "segs": [ { "utf8": "and as I change the distribution for x, it largely stays looking like a bell curve, " } ] }, { "tStartMs": 1368465, "dDurationMs": 3165, "segs": [ { "utf8": "but I can find some distributions that get it to change shape." } ] }, { "tStartMs": 1372230, "dDurationMs": 3644, "segs": [ { "utf8": "For example, the really lopsided one where almost all the probability " } ] }, { "tStartMs": 1375874, "dDurationMs": 3436, "segs": [ { "utf8": "is in the numbers 1 or 6 results in this kind of spiky bell curve." } ] }, { "tStartMs": 1379770, "dDurationMs": 3740, "segs": [ { "utf8": "And if you'll recall, earlier on I actually showed this in the form of a simulation." } ] }, { "tStartMs": 1384130, "dDurationMs": 3999, "segs": [ { "utf8": "Though if you were wondering whether that spikiness was an artifact of the randomness " } ] }, { "tStartMs": 1388129, "dDurationMs": 3721, "segs": [ { "utf8": "or reflected the true distribution, turns out it reflects the true distribution." } ] }, { "tStartMs": 1392290, "dDurationMs": 4180, "segs": [ { "utf8": "In this case, 10 is not a large enough sum for the central limit theorem to kick in." } ] }, { "tStartMs": 1396470, "dDurationMs": 4282, "segs": [ { "utf8": "But if instead I let that sum grow and I consider adding 50 different values, " } ] }, { "tStartMs": 1400752, "dDurationMs": 4667, "segs": [ { "utf8": "which is actually not that big, then no matter how I change the distribution for our " } ] }, { "tStartMs": 1405419, "dDurationMs": 4667, "segs": [ { "utf8": "underlying random variable, it has essentially no effect on the shape of the plot on " } ] }, { "tStartMs": 1410086, "dDurationMs": 604, "segs": [ { "utf8": "the bottom." } ] }, { "tStartMs": 1411170, "dDurationMs": 3665, "segs": [ { "utf8": "No matter where we start, all of the information and nuance for the " } ] }, { "tStartMs": 1414835, "dDurationMs": 3665, "segs": [ { "utf8": "distribution of x gets washed away, and we tend towards this single " } ] }, { "tStartMs": 1418500, "dDurationMs": 3773, "segs": [ { "utf8": "universal shape described by a very elegant function for the standard " } ] }, { "tStartMs": 1422273, "dDurationMs": 4797, "segs": [ { "utf8": "normal distribution, 1 over square root of 2 pi times e to the negative x squared over 2." } ] }, { "tStartMs": 1427810, "dDurationMs": 3000, "segs": [ { "utf8": "This, this right here is what the central limit theorem is all about." } ] }, { "tStartMs": 1431130, "dDurationMs": 4180, "segs": [ { "utf8": "Almost nothing you can do to this initial distribution changes the shape we tend towards." } ] }, { "tStartMs": 1439030, "dDurationMs": 3024, "segs": [ { "utf8": "Now, the more theoretically minded among you might still be " } ] }, { "tStartMs": 1442054, "dDurationMs": 3377, "segs": [ { "utf8": "wondering what is the actual theorem, like what's the mathematical " } ] }, { "tStartMs": 1445431, "dDurationMs": 3479, "segs": [ { "utf8": "statement that could be proved or disproved that we're claiming here." } ] }, { "tStartMs": 1449030, "dDurationMs": 2640, "segs": [ { "utf8": "If you want a nice formal statement, here's how it might go." } ] }, { "tStartMs": 1452130, "dDurationMs": 4510, "segs": [ { "utf8": "Consider this value where we're summing up n different instantiations of our variable, " } ] }, { "tStartMs": 1456640, "dDurationMs": 3577, "segs": [ { "utf8": "but tweaked and tuned so that its mean and standard deviation are 1, " } ] }, { "tStartMs": 1460217, "dDurationMs": 4303, "segs": [ { "utf8": "again meaning you can read it as asking how many standard deviations away from the " } ] }, { "tStartMs": 1464520, "dDurationMs": 830, "segs": [ { "utf8": "mean is the sum." } ] }, { "tStartMs": 1465770, "dDurationMs": 4719, "segs": [ { "utf8": "Then the actual rigorous no-jokes-this-time statement of the central limit theorem " } ] }, { "tStartMs": 1470489, "dDurationMs": 4833, "segs": [ { "utf8": "is that if you consider the probability that this value falls between two given real " } ] }, { "tStartMs": 1475322, "dDurationMs": 4832, "segs": [ { "utf8": "numbers, a and b, and you consider the limit of that probability as the size of your " } ] }, { "tStartMs": 1480154, "dDurationMs": 3980, "segs": [ { "utf8": "sum goes to infinity, then that limit is equal to a certain integral, " } ] }, { "tStartMs": 1484134, "dDurationMs": 4890, "segs": [ { "utf8": "which basically describes the area under a standard normal distribution between those " } ] }, { "tStartMs": 1489024, "dDurationMs": 626, "segs": [ { "utf8": "two values." } ] }, { "tStartMs": 1491250, "dDurationMs": 3896, "segs": [ { "utf8": "Again, there are three underlying assumptions that I have yet to tell you, " } ] }, { "tStartMs": 1495146, "dDurationMs": 2546, "segs": [ { "utf8": "but other than those, in all of its gory detail, " } ] }, { "tStartMs": 1497692, "dDurationMs": 2338, "segs": [ { "utf8": "this right here is the central limit theorem." } ] }, { "tStartMs": 1504550, "dDurationMs": 3442, "segs": [ { "utf8": "All of that is a bit theoretical, so it might be helpful to bring things " } ] }, { "tStartMs": 1507992, "dDurationMs": 4149, "segs": [ { "utf8": "back down to earth and turn back to the concrete example that I mentioned at the start, " } ] }, { "tStartMs": 1512141, "dDurationMs": 3395, "segs": [ { "utf8": "where you imagine rolling a die 100 times, and let's assume it's a fair " } ] }, { "tStartMs": 1515536, "dDurationMs": 2594, "segs": [ { "utf8": "die for this example, and you add together the results." } ] }, { "tStartMs": 1518870, "dDurationMs": 3597, "segs": [ { "utf8": "The challenge for you is to find a range of values such that " } ] }, { "tStartMs": 1522467, "dDurationMs": 3363, "segs": [ { "utf8": "you're 95% sure that the sum will fall within this range." } ] }, { "tStartMs": 1527130, "dDurationMs": 4474, "segs": [ { "utf8": "For questions like this, there's a handy rule of thumb about normal distributions, " } ] }, { "tStartMs": 1531604, "dDurationMs": 4152, "segs": [ { "utf8": "which is that about 68% of your values are going to fall within one standard " } ] }, { "tStartMs": 1535756, "dDurationMs": 3666, "segs": [ { "utf8": "deviation of the mean, 95% of your values, the thing we care about, " } ] }, { "tStartMs": 1539422, "dDurationMs": 2641, "segs": [ { "utf8": "fall within two standard deviations of the mean, " } ] }, { "tStartMs": 1542063, "dDurationMs": 3667, "segs": [ { "utf8": "and a whopping 99.7% of your values will fall within three standard " } ] }, { "tStartMs": 1545730, "dDurationMs": 1240, "segs": [ { "utf8": "deviations of the mean." } ] }, { "tStartMs": 1547450, "dDurationMs": 1978, "segs": [ { "utf8": "It's a rule of thumb that's commonly memorized " } ] }, { "tStartMs": 1549428, "dDurationMs": 2022, "segs": [ { "utf8": "by people who do a lot of probability and stats." } ] }, { "tStartMs": 1552490, "dDurationMs": 2651, "segs": [ { "utf8": "Naturally, this gives us what we need for our example, " } ] }, { "tStartMs": 1555141, "dDurationMs": 2892, "segs": [ { "utf8": "and let me go ahead and draw out what this would look like, " } ] }, { "tStartMs": 1558033, "dDurationMs": 3038, "segs": [ { "utf8": "where I'll show the distribution for a fair die up at the top, " } ] }, { "tStartMs": 1561071, "dDurationMs": 3037, "segs": [ { "utf8": "and the distribution for a sum of 100 such dice on the bottom, " } ] }, { "tStartMs": 1564108, "dDurationMs": 3182, "segs": [ { "utf8": "which by now as you know looks like a certain normal distribution." } ] }, { "tStartMs": 1567950, "dDurationMs": 4705, "segs": [ { "utf8": "Step 1 with a problem like this is to find the mean of your initial distribution, " } ] }, { "tStartMs": 1572655, "dDurationMs": 4877, "segs": [ { "utf8": "which in this case will look like 1 6th times 1 plus 1 6th times 2 on and on and on, " } ] }, { "tStartMs": 1577532, "dDurationMs": 1378, "segs": [ { "utf8": "and works out to be 3.5." } ] }, { "tStartMs": 1579410, "dDurationMs": 4046, "segs": [ { "utf8": "We also need the standard deviation, which requires calculating the variance, " } ] }, { "tStartMs": 1583456, "dDurationMs": 4201, "segs": [ { "utf8": "which as you know involves adding all the squares of the differences between the " } ] }, { "tStartMs": 1587657, "dDurationMs": 2646, "segs": [ { "utf8": "values and the means, and it works out to be 2.92, " } ] }, { "tStartMs": 1590303, "dDurationMs": 2127, "segs": [ { "utf8": "square root of that comes out to be 1.71." } ] }, { "tStartMs": 1592950, "dDurationMs": 2793, "segs": [ { "utf8": "Those are the only two numbers we need, and I will invite you " } ] }, { "tStartMs": 1595743, "dDurationMs": 2793, "segs": [ { "utf8": "again to reflect on how magical it is that those are the only " } ] }, { "tStartMs": 1598536, "dDurationMs": 3154, "segs": [ { "utf8": "two numbers you need to completely understand the bottom distribution." } ] }, { "tStartMs": 1602430, "dDurationMs": 5235, "segs": [ { "utf8": "Its mean will be 100 times mu, which is 350, and its standard deviation " } ] }, { "tStartMs": 1607665, "dDurationMs": 4945, "segs": [ { "utf8": "will be the square root of 100 times sigma, so 10 times sigma, 17.1." } ] }, { "tStartMs": 1613030, "dDurationMs": 4554, "segs": [ { "utf8": "Remembering our handy rule of thumb, we're looking for values two standard " } ] }, { "tStartMs": 1617584, "dDurationMs": 4373, "segs": [ { "utf8": "deviations away from the mean, and when you subtract 2 sigma from mean, " } ] }, { "tStartMs": 1621957, "dDurationMs": 4373, "segs": [ { "utf8": "you end up with about 316, and when you add 2 sigma you end up with 384." } ] }, { "tStartMs": 1627350, "dDurationMs": 1600, "segs": [ { "utf8": "There you go, that gives us the answer." } ] }, { "tStartMs": 1631470, "dDurationMs": 3385, "segs": [ { "utf8": "Okay, I promised to wrap things up shortly, but while we're on this example, " } ] }, { "tStartMs": 1634855, "dDurationMs": 2595, "segs": [ { "utf8": "there's one more question that's worth your time to ponder." } ] }, { "tStartMs": 1638250, "dDurationMs": 2878, "segs": [ { "utf8": "Instead of just asking about the sum of 100 die rolls, " } ] }, { "tStartMs": 1641128, "dDurationMs": 2460, "segs": [ { "utf8": "let's say I had you divide that number by 100, " } ] }, { "tStartMs": 1643588, "dDurationMs": 4502, "segs": [ { "utf8": "which basically means all the numbers in our diagram in the bottom get divided by 100." } ] }, { "tStartMs": 1648570, "dDurationMs": 3000, "segs": [ { "utf8": "Take a moment to interpret what this all would be saying then." } ] }, { "tStartMs": 1652070, "dDurationMs": 5101, "segs": [ { "utf8": "The expression essentially tells you the empirical average for 100 different die rolls, " } ] }, { "tStartMs": 1657171, "dDurationMs": 3768, "segs": [ { "utf8": "and that interval we found is now telling you what range you are " } ] }, { "tStartMs": 1660939, "dDurationMs": 2551, "segs": [ { "utf8": "expecting to see for that empirical average." } ] }, { "tStartMs": 1664350, "dDurationMs": 2693, "segs": [ { "utf8": "In other words, you might expect it to be around 3.5, " } ] }, { "tStartMs": 1667043, "dDurationMs": 3990, "segs": [ { "utf8": "that's the expected value for a die roll, but what's much less obvious and what " } ] }, { "tStartMs": 1671033, "dDurationMs": 3940, "segs": [ { "utf8": "the central limit theorem lets you compute is how close to that expected value " } ] }, { "tStartMs": 1674973, "dDurationMs": 1597, "segs": [ { "utf8": "you'll reasonably find yourself." } ] }, { "tStartMs": 1677590, "dDurationMs": 3101, "segs": [ { "utf8": "In particular, it's worth your time to take a moment mulling over " } ] }, { "tStartMs": 1680691, "dDurationMs": 2773, "segs": [ { "utf8": "what the standard deviation for this empirical average is, " } ] }, { "tStartMs": 1683464, "dDurationMs": 3666, "segs": [ { "utf8": "and what happens to it as you look at a bigger and bigger sample of die rolls." } ] }, { "tStartMs": 1692950, "dDurationMs": 2275, "segs": [ { "utf8": "Lastly, but probably most importantly, let's talk " } ] }, { "tStartMs": 1695225, "dDurationMs": 2185, "segs": [ { "utf8": "about the assumptions that go into this theorem." } ] }, { "tStartMs": 1698010, "dDurationMs": 2306, "segs": [ { "utf8": "The first one is that all of these variables that " } ] }, { "tStartMs": 1700316, "dDurationMs": 2214, "segs": [ { "utf8": "we're adding up are independent from each other." } ] }, { "tStartMs": 1702850, "dDurationMs": 3460, "segs": [ { "utf8": "The outcome of one process doesn't influence the outcome of any other process." } ] }, { "tStartMs": 1707250, "dDurationMs": 3700, "segs": [ { "utf8": "The second is that all of these variables are drawn from the same distribution." } ] }, { "tStartMs": 1711310, "dDurationMs": 3080, "segs": [ { "utf8": "Both of these have been implicitly assumed with our dice example." } ] }, { "tStartMs": 1714790, "dDurationMs": 3597, "segs": [ { "utf8": "We've been treating the outcome of each die roll as independent from the outcome " } ] }, { "tStartMs": 1718387, "dDurationMs": 3643, "segs": [ { "utf8": "of all the others, and we're assuming that each die follows the same distribution." } ] }, { "tStartMs": 1722850, "dDurationMs": 3333, "segs": [ { "utf8": "Sometimes in the literature you'll see these two assumptions lumped " } ] }, { "tStartMs": 1726183, "dDurationMs": 3727, "segs": [ { "utf8": "together under the initials IID for independent and identically distributed." } ] }, { "tStartMs": 1730530, "dDurationMs": 4580, "segs": [ { "utf8": "One situation where these assumptions are decidedly not true would be the Galton board." } ] }, { "tStartMs": 1735710, "dDurationMs": 1120, "segs": [ { "utf8": "I mean, think about it." } ] }, { "tStartMs": 1736970, "dDurationMs": 3207, "segs": [ { "utf8": "Is it the case that the way a ball bounces off of one of the pegs " } ] }, { "tStartMs": 1740177, "dDurationMs": 3013, "segs": [ { "utf8": "is independent from how it's going to bounce off the next peg?" } ] }, { "tStartMs": 1743830, "dDurationMs": 780, "segs": [ { "utf8": "Absolutely not." } ] }, { "tStartMs": 1744770, "dDurationMs": 3100, "segs": [ { "utf8": "Depending on the last bounce, it's coming in with a completely different trajectory." } ] }, { "tStartMs": 1748210, "dDurationMs": 3423, "segs": [ { "utf8": "And is it the case that the distribution of possible outcomes " } ] }, { "tStartMs": 1751633, "dDurationMs": 3037, "segs": [ { "utf8": "off of each peg are the same for each peg that it hits?" } ] }, { "tStartMs": 1755190, "dDurationMs": 1520, "segs": [ { "utf8": "Again, almost certainly not." } ] }, { "tStartMs": 1756710, "dDurationMs": 3523, "segs": [ { "utf8": "Maybe it hits one peg glancing to the left, meaning the outcomes are hugely " } ] }, { "tStartMs": 1760233, "dDurationMs": 3477, "segs": [ { "utf8": "skewed in that direction, and then hits the next one glancing to the right." } ] }, { "tStartMs": 1765730, "dDurationMs": 3441, "segs": [ { "utf8": "When I made all those simplifying assumptions in the opening example, " } ] }, { "tStartMs": 1769171, "dDurationMs": 2459, "segs": [ { "utf8": "it wasn't just to make this easier to think about." } ] }, { "tStartMs": 1771970, "dDurationMs": 2595, "segs": [ { "utf8": "It's also that those assumptions were necessary for this " } ] }, { "tStartMs": 1774565, "dDurationMs": 2505, "segs": [ { "utf8": "to actually be an example of the central limit theorem." } ] }, { "tStartMs": 1778130, "dDurationMs": 3340, "segs": [ { "utf8": "Nevertheless, it seems to be true that for the real Galton board, " } ] }, { "tStartMs": 1781470, "dDurationMs": 4000, "segs": [ { "utf8": "despite violating both of these, a normal distribution does kind of come about?" } ] }, { "tStartMs": 1786050, "dDurationMs": 3944, "segs": [ { "utf8": "Part of the reason might be that there are generalizations of the theorem beyond " } ] }, { "tStartMs": 1789994, "dDurationMs": 3896, "segs": [ { "utf8": "the scope of this video that relax these assumptions, especially the second one." } ] }, { "tStartMs": 1794490, "dDurationMs": 4338, "segs": [ { "utf8": "But I do want to caution you against the fact that many times people seem to assume that " } ] }, { "tStartMs": 1798828, "dDurationMs": 4242, "segs": [ { "utf8": "a variable is normally distributed, even when there's no actual justification to do so." } ] }, { "tStartMs": 1804290, "dDurationMs": 1920, "segs": [ { "utf8": "The third assumption is actually fairly subtle." } ] }, { "tStartMs": 1806210, "dDurationMs": 4060, "segs": [ { "utf8": "It's that the variance we've been computing for these variables is finite." } ] }, { "tStartMs": 1810810, "dDurationMs": 2352, "segs": [ { "utf8": "This was never an issue for the dice example because " } ] }, { "tStartMs": 1813162, "dDurationMs": 1688, "segs": [ { "utf8": "there were only six possible outcomes." } ] }, { "tStartMs": 1815030, "dDurationMs": 3514, "segs": [ { "utf8": "But in certain situations where you have an infinite set of outcomes, " } ] }, { "tStartMs": 1818544, "dDurationMs": 3966, "segs": [ { "utf8": "when you go to compute the variance, the sum ends up diverging off to infinity." } ] }, { "tStartMs": 1823450, "dDurationMs": 3800, "segs": [ { "utf8": "These can be perfectly valid probability distributions, and they do come up in practice." } ] }, { "tStartMs": 1827550, "dDurationMs": 3124, "segs": [ { "utf8": "But in those situations, as you consider adding many different " } ] }, { "tStartMs": 1830674, "dDurationMs": 3571, "segs": [ { "utf8": "instantiations of that variable and letting that sum approach infinity, " } ] }, { "tStartMs": 1834245, "dDurationMs": 3472, "segs": [ { "utf8": "even if the first two assumptions hold, it is very much a possibility " } ] }, { "tStartMs": 1837717, "dDurationMs": 3473, "segs": [ { "utf8": "that the thing you tend towards is not actually a normal distribution." } ] }, { "tStartMs": 1842150, "dDurationMs": 2037, "segs": [ { "utf8": "If you've understood everything up to this point, " } ] }, { "tStartMs": 1844187, "dDurationMs": 3463, "segs": [ { "utf8": "you now have a very strong foundation in what the central limit theorem is all about." } ] }, { "tStartMs": 1848290, "dDurationMs": 3710, "segs": [ { "utf8": "And next up, I'd like to explain why it is that this particular function is the " } ] }, { "tStartMs": 1852000, "dDurationMs": 3990, "segs": [ { "utf8": "thing that we tend towards, and why it has a pi in it, what it has to do with circles." } ] }, { "tStartMs": 1871950, "dDurationMs": 2220, "segs": [ { "utf8": "Thank you." } ] } ] }